Here’s my starting point: the truncated icosahedron, one of the thirteen Archimedean solids.
Next, each face is augmented by a prism, with squares used for the prisms’ lateral faces.
The convex hull of the polyhedron above yields what can be called an expanded truncated icosahedron, as shown below:
Could these faces be made regular, and the polyhedron still hold together? I checked, using Stella 4d‘s “try to make faces regular” function. Here’s the result:
As you can see, the faces of this polyhedron can be made to be regular, but this forces the model to become non-convex.
To try Stella for yourself, for free, just pay a visit to http://www.software3d.com/Stella.php. The trial version is a free download.
RobertLovesPi.net 
I happen to have made a model of that regular polyhedron out of beads recently – there’s a photo on my blog. I was wondering how to make a model in Stella!
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Glad to help! =)
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Your efforts in presenting the POWER OF MATHEMATICS is ABSOLUTELY BRILLIANT
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is it possible to think of the final polyhedron as a great rhombicosidodecahedron augmented with pentagonal cupolae?
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Yes, it is. Good eye!
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It looks like if you remove the protruding part, you will get a polyhedron made by decagon, hexagon, and square.
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That’s correct. The name of this polyhedron is the great rhombicosidodecahedron.
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